A regularized Siegel-Weil formula on \(\text{U}(2,2)\) and \(\text{U}(3)\)
Since the work of Hecke it has been understood that theta series define automorphic forms on classical groups. They allow correspondences between certain pairs of groups, called dual reductive pairs by Howe, the initiator of the spectral approach; however, the idea behind such pairs goes back to the work of \textit{C. L. Siegel} [Ann. Math. (2) 36, 527-606 (1935; Zbl 0012.19703)], which was later reexamined by \textit{A. Weil} [Acta Math. 111, 143-211 (1964; Zbl 0203.03305); Acta Math. 113, 1-87 (1965; Zbl 0161.02304)]. The regularized Siegel-Weil formula for the dual reductive pair \((\text{Sp}(n),O(m))\) was established by \textit{S. Kudla} and \textit{S. Rallis} [Ann. Math. (2) 140, 1-80 (1994; Zbl 0818.11024)]. It relates an integral of theta functions to an Eisenstein series on \(\text{Sp}(n)\). One of the important applications of the regularized Siegel-Weil formula is in studying the relation between the existence of certain poles of the standard \(L\)-functions and the nonvanishing of the corresponding theta liftings. The classical form of this theorem is valid in the range \(m>2n+2\) when the integral on the orthogonal group converges and the Eisenstein series is regular at the point under consideration. Outside the range \(m>2n+2\), the theta integral may not converge and the Eisenstein series may have a pole. S. Kudla and S. Rallis [loc. cit.] have given a method to regularise the theta integral so that the resulting object becomes a meromorphic function. They then established a relation between the first term in the Laurent expansion of the Eisenstein series and the regularised theta integral. In another paper, \textit{S. Kudla, S. Rallis} and \textit{D. Soudry} [Invent. Math. 107, 483-541 (1992; Zbl 0776.11028)] proved a second term identity which involves a comparison between the next terms of the two Laurent series for \(\text{Sp}(2)\) and \(O(4)\). The aim of this paper is to establish the analogous first and second term identities for the dual reductive pair \((U(2,2),U(3))\), which falls outside the convergent range. The methods of this paper essentially follow the general outline of the papers of S. Kudla, S. Rallis and S. Kudla, S. Rallis and D. Soudry cited above. Let \(F\) be a totally real number field and \(E\) a totally imaginary quadratic extension of \(F\). The nontrivial automorphism of \(E\) is denoted by \(a\mapsto\overline{a}\). Let \(\mathbb A\) be the adèle ring of \(F\). Let \(v\) be a place of \(F\). When \(v\) is inert, \(E_v\) is a quadratic extension of \(F_v\). When \(v\) splits as \(\varpi_1\varpi_2\), \(E_v=E_{\varpi_1}\oplus E_{\varpi_2}\) and \(E_{\varpi_i}\simeq F_v\). For an algebraic group \(G\) defined over \(F\), let \(G_v:=G(F_v)\). Assume henceforth that \(G\) is the rank \(n\) quasi-split unitary group \(U(W_{n,n})\) (or \(U(n,n)\) when the underlying space is clear). By fixing a basis, \(W_{n,n}\simeq E^{2n}\) is the split Hermitian space with respect to the Hermitian form \((\;,\;)\) given by the matrix \(\left(\begin{smallmatrix} 0&{I_n}\\ {I_n}&0\end{smallmatrix}\right)\). Let \(W=W_{n,0}\oplus W_{0,n}\) be a complete polarization of \(W_{n,n}\) and \(P\) be the maximal parabolic subgroup of \(G\) which stabilizes \(W_{0,n}\) (usually \(P\) is called the Siegel parabolic subgroup of \(G\)). We have a Levi decomposition of \(P\) into \(MN\), where \[ M:=\left\{m(a)=\begin{pmatrix} a&0\\ 0& {}^t\overline a^{-1}\end{pmatrix}: a\in\text{GL}(n,E)\right\} \] is the Levi factor of \(P\) and \[ N:=\left\{n(b)=\begin{pmatrix} {I_n}&b\\ 0&{I_n}\end{pmatrix}:b\in\text{SH}_n(F)\right\} \] is the unipotent radical of \(P\), where \(\text{SH}_n(F):=\{g\in\text{M}_n(E): {}^t\overline g =-g\}\). Let \(\gamma\) be a Hecke character whose restriction to \(F\) is the quadratic character of the extension \(E/F\) defined by class field theory. One can view \(\gamma\) as a character of \(M(\mathbb A)\) by setting \(\gamma(m(a)):=\gamma(\det(a))\). Then let \(I(s):=\text{Ind}_P^G(\gamma^m|\cdot|_E^s)\) be the normalized induced representation of \(G(\mathbb A)\) obtained by inducing the (quasi-)character \(\gamma^m|\cdot|_E^s\) of \(M(\mathbb A)\) extended trivially to \(N(\mathbb A)\). Its space consists of all smooth functions \(\Phi\) on \(G(\mathbb A)\) such that \[ \Phi(nm(a)f)=\gamma^m(m(a)) |\det(a)|_E^{a+\frac{n}{2}} \Phi(g) \] and \(\Phi\) is \(K_v\)-finite at every archimedean place \(v\). (Here, at the archimedean place \(v\), \(K_v:=U(n)_v\times U(n)_v\).) For each \(v\), one can define similarly \(I_v(s):=\text{Ind}_{P_v}^{G_v}(\gamma_v^m |\cdot|_E^s)\), where \(\gamma_v\) is the local component of \(\gamma\) (that is, a character of \(E_v^\times\)). Allowing \(s\) to vary over the complex plane, one obtains a \textit{section} \(\Phi(s)\in I(s)\). A \textit{standard section} is one whose restriction to \(K\) is independent of \(s\). Given \(\Phi(s)\in I(s)\), the Siegel Eisenstein series \(E(g,s,\Phi)\) is defined as \[ E(g,s,\Phi):=\sum_{\varepsilon\in P(F)\backslash G(F)}\Phi(s)(\varepsilon g). \] This series converges for \(\text{Re}(s)>\frac{n}{2}\) and has a meromorphic continuation to the whole \(s\)-plane. \(E(g,s,\Phi)\) defines an automorphic form on \(G(\mathbb A)\) at each \(s\) where it converges absolutely and when \(\Phi(s)\) is a standard section. Let \(V\) be the skew-Hermitian space of odd dimension \(m\) over \(E\) with skew-Hermitian form \(\langle\;,\;\rangle\) given, with respect to a fixed basis of \(V\), by the \(m\times m\) matrix \((a_{i,j})\), where entries of \((a_{i,j})\) are equal to zero if \(i+j\neq m+1\), and \[ a_{i,j}:=\begin{cases} 1&\text{if \(i<j\)},\\ -1&\text{if \(i>j\)},\\ \xi&\text{if \(i=j\)}, \end{cases} \text{ if }i+j=m+1. \] Here \(\xi\) is trace-zero element in \(E^\times\). Then let \(H\) be the quasi-split unitary group in \(m\) variables \(\text{U}(V)\) (or \(\text{U}(m)\)). Now the dual reductive pair \((G,H)\) can be embedded in \(\text{Sp}(W\otimes_E V)\). Given a nontrivial additive character \(\psi\) of \(\mathbb A/F\), there is a metaplectic representation, depending on \(\psi\), on the metaplectic cover of \(\text{Sp}(W\otimes_E V)(\mathbb A)\). There is an explicit splitting of the metaplectic cover over \(G(\mathbb A)\times H(\mathbb A)\). This splitting is not unique and depends on the choice of \(\psi\) and a Hecke character \(\gamma\) whose restriction to \(F\) is the quadratic character of the extension \(E/F\) defined by class field theory. Then the metaplectic representation induces, via the splitting, an oscillator representation (or Weil representation) \(\omega=\omega(\psi,\gamma)\) of \(G(\mathbb A)\times H(\mathbb A)\). Assume that \(\psi\) and \(\gamma\) have been fixed. The oscillator representation can be realized on \(\mathbb S(W_{n,0}\otimes_E V)=\mathbb S(V(\mathbb A)^n)\), the space of Schwartz functions on \(V(\mathbb A)^n\). This is called the \textit{Schrödinger model} of \(\omega\). One has \(\mathbb S(V(\mathbb A)^n)=\bigotimes_v\mathbb S(V(F_v)^n)\) and each local factor is the model of the local oscillator representation \(\omega_v\). Let \(S(V(\mathbb A)^n)\) denote the subspace of \(\mathbb S(V(\mathbb A)^n)\) such that \(S(V(F_v)^n)\) equals \(\mathbb S(V(F_v)^n)\) for the nonarchimedean place \(v\) and is the subspace of \(\mathbb S(V(F_v)^n)\) corresponding to polynomials in the \textit{Fock model} of \(\omega\) for archimedean \(v\). Let \(\varphi\in S(V(\mathbb A)^n)\) and \((g,h)\in G(\mathbb A)\times H(\mathbb A)\). One defines the theta function \[ \theta(g,h,\varphi):=\sum_{x\in V(F)^n}\omega(g,h) \varphi(x). \] This function is left \(G(F)\)-invariant as a function on \(G(\mathbb A)\). From now on, we assume that \(n=2\) and \(m=3\). Let \[ E(h,s):=\sum_{\varepsilon\in B_H(F)\backslash H(F)}\Phi_s(\varepsilon h), \] where \(\Phi_s(h):=|a(h)|_E^{s+1}\) and \(a(h)=a\) if \(h=m'(a,b)n'k\) is the Iwasawa decomposition of \(h\). (Here \(B_H\) denotes the standard Borel subgroup of \(H\).) This Eisenstein series is absolutely convergent for \(\text{Re}(s)\) large enough and has a meromorphic continuation to the whole complex plane. The author checks that \(\theta(g,h,\omega(z)\varphi)\) is rapidly decreasing on \(H(F)\backslash H(\mathbb A)\) for all \(g\) in \(G\) and \(\varphi\) in \(S(V(\mathbb A)^2)\), and defines the theta integral \[ I(g,s,\omega(z)\varphi):=\int_{H(F)\backslash H(\mathbb A)}\theta(g,h,\omega(z)\varphi) E(h,s) \gamma^{-2} (\det h) dh. \] Then the \textit{regularized theta integral} is defined as \[ {\mathcal E}(g,s,\varphi):={1\over q^s+q^{-s}-q-q^{-1}} I(g,s,\omega(z)\varphi). \] It has at most a double pole at \(s=1\). The Siegel Eisenstein series \(E(g,s,\Phi)\) has at most a simple pole at \(s=\frac{1}{2}\). Let \(A_{-1}(g,\Phi)\) be the leading term in its Laurent expansion at \(s=\frac{1}{2}\), and let \(B_{-2}(g,\varphi)\) be that of \({\mathcal E}(g,s,\varphi)\) at \(s=1\), where \(\Phi\) is the section associated to \(\varphi\) defined by \(\Phi(s,g):=\omega(g) \varphi(0) |a(g)|_E ^{s-\frac{1}{2}}\). The author proves the following: If \(B_{-2}(g,\varphi)\) does not vanish, then \(A_{-1}(g,\Phi)=cB_{-2}(g,\varphi)\) for some nonzero constant \(c\). Now let \(A_0(g,\Phi)\) and \(B_{-1}(g,\varphi)\) be the respective second terms in the Laurent expansion of \(E(g,s,\Phi)\) and \({\mathcal E}(g,s,\varphi)\). The following holds: \(A_0(g,\Phi)=c'B_{-1}(g,\varphi)+\Psi(g)\) for some \(\Psi\in A_{-1}(g,\Phi(\frac{1}{2}))\) and some nonzero constant \(c'\).
analytic continuation of Eisenstein series
scientific article; zbMATH DE number 1425144
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